Problem 43 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(1-\frac{x}{2}>4\)

Step-by-Step Solution

Answer

The solution to the inequality \(1-\frac{x}{2}>4\) in interval notation is \((-∞, -6)\).

Step 1: Isolate the Variable

To start, subtract 1 from both sides of the inequality and then multiply both sides by -2 to isolate x.\n\n\[1-\frac{x}{2}>4\] becomes \[\frac{x}{2}< -3\] and finally multiplying both side by 2 we get \(x<-6\).

Step 2: Express in Interval Notation

In this step, the inequality \(x<-6\) will be written in interval notation (a, b) where 'a' and 'b' are endpoints of the interval. This inequality can be expressed as \((-∞, -6)\). The parenthesis signifies that -6 is not included in the solutions.

Step 3: Draw on a Number Line

Finally, the solution set will be graphically represented on a number line. A open circle will be placed above -6 (because -6 is not included in solution set) and a line will be drawn extending to the left towards negative infinity. This visually represents all the possible x-values that satisfy the inequality.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It conveniently describes the set of solutions by using intervals. In our problem, we have the inequality \(x < -6\). To express this using interval notation, we think about all the numbers that are less than -6. These numbers stretch all the way to negative infinity, hence we write the interval as \((-\infty, -6)\).

• The round bracket, \(-\infty\), means that negative infinity is not a number and cannot be reached.
• The round bracket, \(-6\), indicates that -6 is not included in the set of solutions (as opposed to a square bracket \([-6]\), which would include -6).

Writing inequalities in interval notation is convenient especially for solutions sets of linear inequalities, as it offers a neat and standardized way to represent ranges of solutions. Remember to use round brackets for exclusive intervals and square brackets for inclusive intervals.

Number Line Graphing

Number line graphing is a visual method used to represent the solution set of an inequality. For example, with our inequality solution \(x < -6\), we need to visually show all possible values of \(x\) that are less than -6. Here's how you do it:

• First, draw a horizontal line to represent the number line.
• Identify where -6 should be on this line and place an open circle on that point to indicate that -6 itself is not included in the solution set.
• Draw a line extending to the left from -6 towards negative infinity to show that all values less than -6 are part of the solution.

This shading (or line) helps anyone viewing the number line see precisely which values \(x\) can take for the inequality to hold true.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that satisfy an inequality. Let's break down the process using our example inequality: \(1 - \frac{x}{2} > 4\).

• The first step is to isolate the variable, which often involves reversing operations — similar to solving equations. For our example, we subtract 1 from both sides to simplify.
• Next, when multiplying or dividing by a negative number, it's crucial to remember that the inequality sign reverses. This is why our inequality transforms from \(\frac{x}{2} < - 3\) after subtracting and multiplying terms appropriately.
• Completing the solution involves performing operations that completely isolate \(x\). In this case, multiplying both sides by 2 results in \(x < -6\).

Remember, a clear and methodical process is essential in solving inequalities to ensure that each step is done correctly, especially when handling negative numbers affecting the direction of the inequality.